Tangential and Normal Components of Acceleration

Definition

$\mathbf{a}={\mathbf{r}}^{\prime \prime }=a_T\mathbf{T}+a_N\mathbf{N}$ where $a_T=\frac{d^{2}s}{d{t}^2}=\frac{d}{dt}|{\mathbf{r}}^{\prime }(t)|$ and $a_N=\kappa {\left(\frac{ds}{dt}\right)}^2=\kappa |{\mathbf{r}}^{\prime }(t){|}^2=\sqrt{|{\mathbf{r}}^{\prime \prime }{|}^2+a_T^2}$